Superlattice patterns and spatial instability induced by delay feedback

Abstract

Various oscillatory superlattice patterns in a reaction–diffusion system are observed by means of delay feedback (DF) in the parametric domain where the system without DF displays uniform bulk oscillation. By varying DF parameters within an appropriate range, the system undergoes transitions to oscillatory hexagons, stripes and squares, and square superlattices with different wavenumbers are also obtained. Linear stability analysis reveals that the patterns do not result from the Turing instability and a possible mechanism of pattern formation is suggested and proved analytically: DF induces instability of a homogeneous limit cycle with respect to spatial perturbations even if the Turing instability does not occur, so that oscillatory patterns possessing the corresponding spatial modes are produced. The different behavior of the dominant characteristic multiplier seems to be connected to the pattern selection. Here it is clearly demonstrated that DF can play a destabilizing role in spatially extended system instead of stabilizing the periodic orbits or turbulent states, which most earlier works have usually focused on.